Degree Associated Reconstruction Parameters of Total Graphs

A card (ecard) of a graph G is a subgraph formed by deleting a vertex (an edge). A dacard (da-ecard) specifies the degree of the deleted vertex (edge) along with the card (ecard). The degree associated reconstruction number (degree associated edge reconstruction number ) of a graph G, drn(G) (dern(G)), is the minimum number of dacards (da-ecards) that uniquely determines G. In this paper, we investigate these two parameters for the total graph of certain standard graphs

Regular configurations and TBR graphs

PhD 2009 QMThis thesis consists of two parts: The first one is concerned with the theory and applications of regular configurations; the second one is devoted to TBR graphs. In the first part, a new approach is proposed to study regular configurations, an extremal arrangement of necklaces formed by a given number of red beads and black beads. We first show that this concept is closely related to several other concepts studied in the literature, such as balanced words, maximally even sets, and the ground states in the Kawasaki-Ising model. Then we apply regular configurations to solve the (vertex) cycle packing problem for shift digraphs, a family of Cayley digraphs. TBR is one of widely used tree rearrangement operationes, and plays an important role in heuristic algorithms for phylogenetic tree reconstruction. In the second part of this thesis we study various properties of TBR graphs, a family of graphs associated with the TBR operation. To investigate the degree distribution of the TBR graphs, we also study -index, a concept introduced to measure the shape of trees. As an interesting by-product, we obtain a structural characterization of good trees, a well-known family of trees that generalizes the complete binary trees

Local-set-based Graph Signal Reconstruction

Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios.Comment: 28 pages, 9 figures, 6 tables, journal manuscrip